Robust Computation in Engineering, Geometry and Duality ICONS

Vaclav Skala http://www.VaclavSkala.eu ICONS 2012 1

Robust Computation in Engineering, Geometry and Duality

ICONS 2012 Reunion Island

Courtesy of http://www.google.cz

Václav Skala

University of West Bohemia, Plzen, Czech Republic VSB‐Technical University, Ostrava, Czech Republic

http://www.VaclavSkala.eu


Plzen (Pilsen) City

Plzen is an old city [first records of Plzen castle 976] city of culture, industry, and brewery.

City, where today’s beer fermentation process was invented that is why today’s beers are called Pilsner ‐ world wide


Ostrava City

Ostrava is

• an industrial city of coal mining & iron melting

• 3rd largest city


University of West Bohemia 17530 students + 987 PhD students

Computer Science and Engineering Mathematics (including Geomatics)

Physics Cybernetics Mechanics (Computational)

• Over 50% of income from research and application projects • NTIS project (investment of 27 mil. EUR) • 2nd in the ranking of Czech technical / informatics faculties 2009, 2012


“Real science” in the XXI century

Courtesy of Czech Film, Barrandov


History of Mathematics

• Euclid ‐ synthetic geometry 300 BC • Descartes ‐ analytic geometry 1637 • Gauss – complex algebra 1798 • Hamilton – quaternions 1843 • Grassmann – Grasmann Algebra 1844 • Cayley – Matrix Algebra 1854 • Clifford – Clifford algebra 1878 • Gibbs – vector calculus 1881 • Sylvester – determinants 1878 • Ricci – tensor calculus 1890 • Cartan – differential forms 1908 • Dirac, Pauli – spin algebra 1928

Hestenes – Space‐time algebra 1966 Geometry Algebra 1984

Main line


Robust Computation in Engineering, Geometry and Duality

• Typical engineering and geometrical problems

• Euclidean and projective spaces

• Duality, property of dual transformation

• Algorithm complexity and dual problems

• Robustness & influence to algorithm design

• Typical examples of duality applications


Numerical systems

• Binary system is used nearly exclusively • Octal & hexadecimal representation is used • If we would be direct descendants of tetrapods – we would have a great advantage – “simple counting”

Name Base Digits E min E max BINARY

B 16 Half 2 10+1 −14 15B 32 Single 2 23+1 −126 127

B 64 Double 2 52+1 −1022 1023 B 128 Quad 2 112+1 −16382 16383

DECIMALD 32 10 7 −95 96D 64 10 16 −383 384D 128

10 34 −6143 6144

IEEE 758‐2008 standard

Figure 1


Mathematically perfect algorithms fail due to instability

• Main issues

– stability, robustness of algorithms

– acceptable speed

– linear speedup – results depends on HW, CPU …. parameters !

• Numerical stability

– limited precision of float / double

– tests A ? B with floats

• if A = B then ….. else ….. ; if A = 0 then ….. else …. should be forbidden in programming languages

– division operation should be removed or postponed to the last moment if possible ‐ “blue screens”, system resets

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Vectors and Points in Geometry

Vector representation v = ( vx , vy , vz : 0 )

Point representation P = ( Px , Py , Pz : 1 ), resp. ( Px , Py , Pz : Pw ),

Many libraries do not distinguish between points and vectors and treat them in the same manner !! BE CAREFUL !!

v = P1 – P0 = ( Px1 , Py1 , Pz1 : 1 ) ‐ ( Px0 , Py0 , Pz0 : 1 ) = ( Px1 – Px0 , Py1 – Py0 , Pz1 – Pz0 : 0 ) = ( vx , vy , vz : 0 )

!!! Do not make it on CPU/GPU – result ( vx , vy , vz : ε )

It is a point ( vx / ε, vy /ε, vz / ε ) in E3


Floating point

• Not all numbers are represented correctly

• Logarithmic arithmetic • Continuous fractions • Interval arithmetic

• NOT valid identities

1

x y

4

1 1

3 2

5 3…

x + y = [a + c, b + d] x = [ a , b ]

x ‐ y = [a ‐ d, b ‐ c] y = [ c , d ]

x × y = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]

x / y = [min(a/c, a/d, b/c, b/d),

max(a/c, a/d, b/c, b/d)] if y ≠ 0

3; 7,15,1,292,1,1,1,2,1,3,1 …


Statements like

if <float> = <float> then …. or if <float> ≠ <float> then ….

should not be

Quadratic equation

0

If 4 then

4 /2

⁄

to get more reliable results.


Function value computation at 77617, 33096

, 333.75 11 121 2 5.5 / 2

6.33835 10 single precision

1,1726039400532 double precision

1,1726039400531786318588349045201838 extended precision

The correct result is “somewhere” in the interval of

0,827396059946821368141165095479816292005, 0,827396059946821368141165095479816291986

Exact solution

, 2 25476766192


Summation is one of often used computations.

10 0.999990701675415

or

10 1.000053524971008

The result should be one. The correctness in summation is very important in power series computations. !!!!ORDER of summation

1 14.357357

114.392651


Recursion

• Towers o Hanoi

• Ackermann function

The value of the function grows very fast as

4,4 2 2

MOVE (A, C, n); MOVE (A, B, n‐1); MOVE (A, C, 1); MOVE (B, C, n‐1) # MOVE (from, to, number) #

,1 0

1,1 0 01, , 1 0 0


Numerical computations

Hilbert’s Matrix

1 1 1 1 21

1.0E-131.0E-111.0E-091.0E-071.0E-051.0E-031.0E-011.0E+011.0E+031.0E+051.0E+071.0E+091.0E+111.0E+131.0E+151.0E+171.0E+19

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Err

or

Order of the Hilbert matrix

ε

εp

ξ


Mathematical “forms” There are several “forms”:

• Implicit , , 0 0 there is no orientation, e.g.

o if 0 is a iso‐curve there is no hint how to find another point of this curve, resp. a line segment approximating the curve => tracing algorithms

o if 0 is a iso‐surface there is no hint how to find another point of this surface => iso‐surface extraction algorithms

• Parametrical – , points of a curve are “ORDERED” according to a parameter

• Explicit , for the given values , resp. , we get function value

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Data processing ‐ main field in computer science

Data processing itself can be split to two main areas:

• processing of textual data limited interval of values, unlimited dimensionality

[char as one dimension ‐

Methionylthreonylthreonylglutaminylarginyl...isoleucine 189,819 chars] ‐ no interpolation is defined

• processing of numerical data unlimited interval of values, limited dimensionality – usually 2 or 3

‐ interpolation can be used

Textual Graphical

Dim ∞ 2, 3

Interval 0‐255 (ASCII)

(‐∞, ∞)


Hash functions

• usually used for textual data processing • prime numbers and modulo operations are used for the hash function

Usual form

3 5 7

multiplication int * float is needed 271

11

10

100

1000

1 2001 4001 6001 8001 10001 12001

Num

ber o

f buc

kets

Bucket length

18

1

10

100

1 201 401 601 801 1001 1201

Num

ber o

f buc

kets

Bucket length


If the hash function is constructed as

where , , are “irrational” numbers and 2 1 better distribution is

obtained => much faster processing.

173878

1

10

100

1000

10000

100000

1000000

1 10 100 1000

Num

ber o

f clu

ster

s

Cluster length

data set: A4_unterbau1.stl


Textual processing

The has function is constructed as

„irrational“ 0 1

2 1

Both geometrical and textual hash function design have the same approach coefficients are “irrational” and no division operation is needed.

Some differences for Czech, Hebrew, English, German, Arabic, … languages and “chemical” words.

1

10

100

1000

10000

0 0.2 0.4 0.6 0.8 1

q

English dictionary (start)


Projective Space

X = [X, Y]T X∈E2

x = [ x, y: w]T x∈P2

Conversion:

X = x / w Y = y / w

& w ≠ 0

If w = 0 then x represents “an ideal point” ‐ a point in infinity, i.e. it is a directional vector.

The Euclidean space E2 is represented as a plane w = 1.

x y

w

w=1x

X Y

(a)

pP2

E2

ρ

a b

c

c=1D(p)

D( )ρ

A B

(b)

D(P )2

D(E ) 2


Duality

For simplicity, let us consider a line p defined as:

aX + bY + c = 0

We can multiply it by w ≠ 0 and we get:

ax + by + cw = 0 i.e. pTx = 0 p = [ a, b: c]T x = [ x, y: w]T

It is actually a plane in the projective space P2 (point [0, 0: 0]T excluded),

i.e. line p∈E2: p = [a, b: c]T

x y

w

w=1x

X Y

(a)

pP2

E2

ρ

a b

c

c=1D(p)

D( )ρ

A B

(b)

D(P )2

D(E ) 2


From the mathematical notation pTx = 0

we cannot distinguish whether p is a line and x is a point or vice versa in the case of P2. It means that a point and a line are dual in the case of P2, and a point and a plane are dual in the case of P3.

The principle of duality in P2 states that:

Any theorem remains true when we interchange the words “point” and “line”, “lie on” and “pass through”, “join” and “intersection”, “collinear” and “concurrent” and so on.

Once the theorem has been established, the dual theorem is obtained as described above.

This helps a lot to solve some geometrical problems.


Definition 1

The cross product of the two vectors x1 = [x1,y1:w1]T and x2 = [x2,y2:w2]

T is defined as:

1 2 1 1 1

2 2 2

× det x y wx y w

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

i j kx x

where: i = [1,0,0]T, j = [0,1,0]T, k = [0,0,1]T

Please, note that homogeneous coordinates are used.


Theorem 1

Let two points x1 and x2 be given in the projective space. Then the coefficients of the p line, which is defined by those two points, are determined as the cross product of their homogeneous coordinates

p = x1× x2

Proof 1

Let the line p∈E2 be defined in homogeneous coordinates as

ax + by + cw = 0

We are actually looking for a solution to the following equations:

1 0T =p x and 2 0T =p x where: p = [a,b,c]T


It means that any point x that lies on the p line must satisfy both the

equation above and the equation 0T =p x in other words the p vector is

defined as

1 1 1

2 2 2

det x y wx y w

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

i j kp

We can write ( )1 2× 0T =x x x

i.e.

1 1 1

2 2 2

det 0x y wx y wx y w

⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦


Evaluating the determinant

0

we get the line coefficients of the line p as:

⎥⎦

⎤⎢⎣

⎡=

22

11detwywy

a

⎥⎦

⎤⎢⎣

⎡−=

22

11detwxwx

b

⎥⎦

⎤⎢⎣

⎡=

22

11detyxyx

c

Note: for w = 1 we get the standard cross product formula and the cross product defines the p line, i.e. p = x1 × x2

where: p = [a,b:c]T

• An intersection of two lines => A x = b • A line given by two points => A x = 0

Cross product is equivalent to a solution of a linear system of

equations !

No division operation !


Computation in Projective Space

• Cross product definition

• A plane ρ is determined as a cross product of three given points

Due to the duality

• An intersection point x of three planes is determined as a cross product of three given planes

Computation of generalized cross product is equivalent to a solution of a linear system of equations => no division operation!

1 1 1 11 2 3

2 2 2 2

3 3 3 3

× ×x y z wx y z wx y z w

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

i j k l

x x x

1 1 1 11 2 3

2 2 2 2

3 3 3 3

× ×a b c da b c da b c d

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

i j k l

ρ ρ ρ



E2 E3

Dual problem

In graphical applications position of points are changed by an interaction, i.e. . The question is how coefficients of a line, resp. a plane are changed without need to recompute them from the definition.

It can be proved that , resp.

As the computation is made in the projective space we can write

, resp.

THIS IS SIGNIFICANT SIMPLIFICATION OF COMPUTATIONS

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Transformation about a general axis

1

1 where:

Instead of usually used transformations:

That is generally complex, unstable as a user has to select which axis is to be used for a rotation



Interpolation

• It means that we can interpolate using homogeneous coordinates without a need of “normalization” to Ek !!

• Homogeneous coordinate w ≥ 0 In many algorithms, we need “monotonous” parameterization, only

( )

( )

0 1 0

0 1 0

0 1 0 0 1 0

0 1 0 0 1 0

Linear parametrization( ) ( ) ,

Non-linear (monotonous) parametrization( ) ( ) ,( ) ( ) ( ) ( )( ) ( ) ( ) ( )

t t t

t t tx t x x x t y t y y y tz t z z z t w t w w w t

= + − ∈ −∞ ∞

= + − ∈ −∞ ∞

= + − = + −

= + − = + −

X X X X

x x x x

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.0 0.2 0.4 0.6 0.8 1.0 1.2

1.50

1.70

1.90

2.10

2.30

2.50

2.70

2.90

3.10

0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00

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Spherical interpolation

, ,sin 1 Ω

sin Ωsin Ω

sin Ω

Instability occurs if Ω .

Mathematically formula is correct,

in practice the code generally incorrect! [ ]

, , sin 1 Ω sin Ωsin Ω

sin 1 Ω sin Ω : sin Ω

Homogeneous coordinate



Intersection line – plane

An intersection of a plane with a line in E2 / E3 can be computed efficiently

Comparison operations must be modified !!!

Cyrus‐Beck line clipping algorithm 10‐25% faster

( )

[ ]

0 1 0

1 0

0

0

min

min min

( ) ( ) ,

0 0

, , ,

: 0 :

TEST.....

* * ..... condition 0w

T

T

T

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T Tw

w w

w

t t t

ax by cz d

a b c d

t

t t

τ τ

τ τ τ

τ τ τ τ τ

= + − ∈ −∞ ∞

= + + + =

= = −

= −

= − =

= ≤ = −⎡ ⎤⎣ ⎦

>> ≥

x x x x

a x

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a xa sa x a s

t if then t t

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Computation in Projective Space ‐ Barycentric coordinates

Let us consider a triangle with vertices X1, X2, X3,

A position of any point X∈E2 can be expressed as

Linear system must be solved

If points xi are given as [xi, yi, zi: wi ]T and wi ≠ 1then xi must be “normalized”

1 1 2 2 3 3

1 1 2 2 3 3

1 2 3

additional condition1 0 1

i = 1,...,3

i

ii

a X a X a X Xa Y a Y a Y Y

a a a aP

aP

+ + =

+ + =

+ + = ≤ ≤

=

P1

x1

x

x3

P3

x2

P2



Rewriting

11 2 3

21 2 3

3

41 1 1 1

bX X X X

bY Y Y Y

bb

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦

0

Solution of the linear system of equations (LSE) is equivalent to generalized cross product

1 1 2 2 3 3 4

1 1 2 2 3 3 4

1 2 3 4

4 4

00

01,...,3 0i i

b X b X b X b Xb Y b Y b Y b Yb b b bb a b i b

+ + + =

+ + + =

+ + + =

= − = ≠

[ ]

1 2 3 4

1 2 3

1 2 3

× ×

, , ,

, , ,

, , ,

1,1,1,1

T

T

T

T

b b b b

X X X X

Y Y Y Y

=

= ⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

=

b ξ η w

b

ξ

η

w



if wi ≠ 1

=> entities: projective scalar, projective vector

( Skala,V.: Barycentric coordinates computation in homogeneous coordinates, Computers&Graphics, 2008)

11 2 3

21 2 3

31 2 3

4

bx x x x

by y y y

bw w w w

b

⎡ ⎤⎡ ⎤ ⎢ ⎥⎢ ⎥ ⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦

0

1 2 3 4

1 2 3

1 2 3

1 2 3

× ×

, , ,

, , ,

, , ,

, , ,

T

T

T

T

b b b b

x x x x

y y y y

w w w w

=

= ⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

= ⎡ ⎤⎣ ⎦

b ξ η w

b

ξ

η

w

1 2 3

2 3 1

3 1 2

0 ( : ) 10 ( : ) 10 ( : ) 1

b w w wb w w wb w w w

≤ − ≤

≤ − ≤

≤ − ≤

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Standard formula

The formula is quite “horrible” one and for students not acceptable as it is too complex and they do not see from the formula comes from.



, , : , , :

normal vectors are

, , , ,

directional vector of a line of two planes and is given as

“starting” point ???

A plane passing the origin with a normal vector , , , : 0

The point is defined as

How simple formula supporting matrix‐vector architectures like GPU and parallel processing



Advantages

• “infinity” is well represented • No division operation is needed • Many mathematical formula are simpler • One code sequence solve primary and dual problems • Supports matrix – vector operations in hardware – like GPU etc.

Disadvantages

• Careful handling with formula as the projective space • Exponents of the homogeneous vector can overflow – it should be normalized not supported by the current hardware – P_Lib use on GPU (C# and C++)

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4D cross product can be implemented in Cg/HLSL on GPU (not optimal implementation) as:

float4 cross_4D(float4 x1, float4 x2, float4 x3) float4 a; a.x=dot(x1.yzw, cross(x2.yzw, x3.yzw)); a.y=‐dot(x1.xzw, cross(x2.xzw, x3.xzw)); a.z=dot(x1.xyw, cross(x2.xyw, x3.xyw)); a.w=‐dot(x1.xyz, cross(x2.xyz, x3.xyz)); return a;


Acknowledgment Questions

Thanks belong to colleagues at UWB & VSB universities for discussions and help, to many authors of published related papers, I haven’t refer properly (a list would be too long*). Research supported by the MSMT CR, projects No. LA10035, ME10060

Contact: Vaclav Skala http://www.VaclavSkala.eu University of West Bohemia, Plzen & VSB‐Technical University, Ostrava Czech Republic *) Please, find related reference in: • Skala,V.: Geometry, Duality and Robust Computation in Engineering, submitted for publication, 2012

• Skala,V.: Interpolation and Intersection Algorithms and GPU, ICONS2012, VisGra 2012 workshop, accepted for publications, 2012


Main References [1] van Verth,J.M., Bishop,L.M.: Essential Mathematics for Games and Interactive

Applications, Morgan Kaufmann 2005 [2] Skala,V.: GPU Computation in Projective Space Using Homogeneous Coordinates ,

Game Programming GEMS 6 (Ed.Dickheiser,M.), pp.137‐147, Charles River Media, 2006 [3] Skala,V.: A new approach to line and line segment clipping in homogeneous

coordinates, The Visual Computer, Vol.21, No.11, pp.905‐914, Springer Verlag, 2005 [4] Skala,V.: Barycentric coordinates computation in homogeneous coordinates, submitted

for publication, 2007 [5] Skala,V.: Length, Area and Volume Computation in Homogeneous Coordinates,

International Journal of Image and Graphics, Vol.6., No.4, pp.625‐639, 2006 [6] Yamaguchi,F.: Computer Aided Design – A totally Four‐Dimensional Approach, Springer

Verlag 2002 [7] Fernando,R., Kilgard,M.J.: The Cg Tutorial, Addison Wesley, 2003 [8] Skala,V., Kaiser,J., Ondracka,V.: Library for Computation in the Projective Space,

Aplimat 2007 conf., 2007 [9] Uhlir,K., Skala,V.: Radial Basis Functions, EUSIPCO 2005


RESOURCES FOR ROBUST COMPUTATION Numerical nonrobustness causes all kinds of failures. But can you produce an example with an infinite loop? This and other forms of manifestation are discussed in Anatomy of Algorithmic Failures. It is intended to provide classroom examples. Source code available. Robust Geometric Algorithms and their Implementation, Guest Editorial Forward from a Special Issue of Computational Geometry: Theory and Applications (CGTA 33:1, 2006). Resource Page for ``Survey/Tutorial on Exact Geometric Computing'' Lectures at Workshop on Geometric Computing, University of Hong Kong, June 27‐‐29, 2001.

General Forums on Nonrobustness Issues ‐‐ DIMACS Workshop on Implementation of Geometric Algorithms Dec 4‐6, 2002 ‐‐ SIAM Minisymposium: Robust Geometric Computation (2001) ‐‐ MSRI Workshop on Foundations of CAD (1999) ‐‐ SIAM Workshop on Integration of CAD and CFD (1999) ‐‐ Emerging Challenges in Computational Topology Report (1999) ‐‐Computational Geometry Task Force Report (1996) ‐‐ SIGGRAPH'98 Panel on Robust Geometry. Position Statements from panelists. ‐‐ACM Strategic Directions Report (1996) ‐‐Tom Peters's presentation on Non‐robustness Issues in CAGD [here is a local copy] This work is part of the NSF/DARPA CARGO Program (2001‐4) in which non‐robustness and topological consistency issues are addressed.

Non‐robustness in the News ‐‐ Patriot Missile Defense Saga, from US General Accounting Office (local copy) ‐‐ Ariane 5 Saga, from European Space Agency ‐‐ Software Bugs Cost US economy $59.5 billion/year. The report focused on the financial sector, and the automotive and aircraft manufacturing industries. In the latter 2 industries, the cost is estimated at $1.8 billion/year. Report was prepared by Research Triangle Institute for NIST. ‐‐ Disasters attributable to numerical errors, from Doug Arnold (including the North Sea oil rig collapse)

Projects and Groups ‐‐K. Mehlhorn at Max‐Planck Institute of Computer Science is involved in various robustness projects (LEDA, CGAL, EXACUS). ‐‐PRISME Project at INRIA, Sophia‐Antipolia, directed by J.‐D. Boissonnat. ‐‐ Superrobust Computation Project of K. Sugihara at Tokyo University ‐‐ Arenaire Project of J.‐M. Muller. ‐‐ iRRAM Project of N. Mueller is a C++ package for error‐free real arithmetic based on the concept of a real RAM. ‐‐ Professor Cuyt's Group on Computer Arithmetic and Numerical Techniques. ‐‐ Evaluation of Special Functions from Dan Lozier at NIST: includes a function evaluator service.

RETRIEVED from http://www.cs.nyu.edu/csweb/Research/Groups/ http://www.cs.nyu.edu/csweb/Research/Groups/


Software ‐‐ GMP Home. New in GMP 3.1 (FFT Based Multiplication!) ‐‐MPFR homepage The MPFR library is a C library for multiprecision floating‐point computations with exact rounding. It is based on the GMP multiprecision library and will replace the MPF class starting with version 3.1 of GMP. Here are some timings . ‐‐ Intel's Open source for Numerics ‐‐ Arithmetic Explorer

Robust Meshing and Triangulation ‐‐ GNU Triangulated Surface (GTS) Library: Open Source free software for 3D surfaces triangular meshes. Version 0.4.0 (Jan 2001). FEATURES: based on the GTK+ GUI Toolkit, 2D dynamic constrained Delaunay triangulation, robust predicates of Shewchuk, Boolean set operations, multiresolution models, dynamic view‐independent LOD, some view‐dependent LOD, Kd‐trees, collision detection. [ Bibliography | Reference Manual | ]

Computing transcendental functions. ‐‐ The Table Maker's Dilemma ‐‐ Elementary Functions, Algorithms and Implementation, book by Jean‐Michel Muller from Birkhauser, Boston (1997).

Organizations and Web Resources ‐‐ CCA Net (Computability and Complexity in Analysis) ‐‐ Interval Computation ‐‐ Real Algebraic and Analytic Geometry (RAAG) ‐‐ Foundations of Computational Mathematics

Software Resources Source Forge: free Open Source developer services (CVS, mailists, forums, site hosting, etc). Thousands of projects in every field.

Conferences ‐‐ IEEE Symposium on Computer Arithmetic. Call for Papers for 15th Symposium on Computer Arithmetic (Nov 1, 2000). ‐‐ 4th Conference on Real Numbers and Computers ‐‐ 4th Workshop on Computability and Complexity in Analysis Sep 17‐20, 2000, Swansea, Wales.

Bibliography Collection ‐‐ Constructivity, Computability and Complexity in Analysis, a collection of over 500 entries. ‐‐ Numerical techniques from Rational Approximation Theory and Computer Arithmetic, a collection of over 2500 entries.

RETRIEVED from http://www.cs.nyu.edu/csweb/Research/Groups/ http://www.cs.nyu.edu/csweb/Research/Groups/

Documents

Robust Computation in Engineering, Geometry and Duality ICONS